Ricci solitons with an orthogonally intransitive 2-dimensional Abelian Killing algebra

TL;DR

This paper classifies four-dimensional Ricci solitons with a specific 2-dimensional Abelian symmetry algebra, under certain geometric conditions, contributing new examples and understanding to the study of orthogonally intransitive Ricci solitons.

Contribution

It provides a local classification of Ricci solitons with a 2D Abelian Killing algebra under specific null and transitivity conditions, filling a gap in the literature.

Findings

Classification of Ricci solitons with specified symmetry properties

Identification of conditions for null curvature vector fields

Examples of orthogonally intransitive Ricci solitons

Abstract

In this paper we report on a local classification of four dimensional Ricci solitons which have a $2$-dimensional Abelian Killing algebra $\mathcal{G}_{2}$, whose Killing leaves are non-null and orthogonally intransitive. The classification is obtained under the following additional assumptions: (i) the curvature vector field, of the submersion defined by $\mathcal{G}_{2}$, is a null vector field; (ii) $\mathcal{G}_{2}$ has a null vector; (iii) the vector field of the Ricci soliton is tangent to the Killing leaves and a symmetry of the orthogonal distribution. Since there are only few examples of orthogonally intransitive Einstein metrics, and even less is known about orthogonally intransitive Ricci solitons, we believe that these results can help fill this gap in the literature.